Question

Prove that:
Opposite angles of a cyclic quadrilateral
are supplementary.why​

Answer 1

Given :- O is the centre of circle. ABCD is the cyclic quadrilateral.

To prove :- ∠BAD + ∠BCD  =  180°, ∠ABC + ∠ADC  =  180°

Construction : Join OB and OD

Proof :-

=> (i) ∠BAD  =  (1/2)∠BOD.

(The angle substended by an arc at the centre is double the angle on the circle.)

=> (ii) ∠BCD  =  (1/2) reflex ∠BOD.

=> (iii) ∠BAD + ∠BCD  =  (1/2)∠BOD + (1/2) reflex ∠BOD.

=> Add (i) and (ii).

∠BAD + ∠BCD  =  (1/2)(∠BOD + reflex ∠BOD)

∠BAD + ∠BCD  =  (1/2) ⋅ (360°)

(Complete angle at the centre is 360°)

∠BAD + ∠BCD  =  180°

=> (iv) Similarly ∠ABC + ∠ADC  =  180°.

I hope this will help you dear..

Always stay safe and stay healthy..